Fundamental Matrix
1. Weak Calibration
We have two cameras but don't know their calibration parameters (such as focal length).
So need to estimate the epipolar geometry from a redundant set of point correspondences across the uncalibrated cameras.
2. Calibration Matrix
Recall that the calibration matrix is: For convenience sake, we'll assume that there is no skew . This makes the intrinsic parameter matrix invertible: Recall that the extrinsic parameter matrix is what maps points from world space to points in the camera's coordinate frame, meaning:
Since we said that the intrinsic matrix is invertible, that also means that: Which means that we can find a ray through the camera and the world (since it's a homogeneous point in 2-space, and recall point-line duality) corresponding to this points.
3. Fundamental matrix constraint
For two cameras:
Now note we don't know the values of for either camera since we are working in the uncalibrated case, but we do know that there are some parameters that would calibrate them.
There is a well-defined relationship between the left and right points in the calibrated case using essential matrix: Thus, via substitution, we have: After rearrangement: This gives us: OR simpler:
3.1 Properties of the Fundamental Matrix
TBD